了解Scala中的monad

Question

I'm trying to understand what monads are (not just in scala, but by example using scala). Let's consider the most (in my opinion) simple example of a monad:

scala.Some

As some articles state, every monad in its classic sense should preserve some rules for the flatMap and unit functions.

Here is the definition from scala.Some

@inline final def flatMap[B](f: A => Option[B]): Option[B]

So, understand it better I want to understand it from the category theory standpoint . So, we're considering a monad and it's supposed to be a functor ( but between what? ).

Here we have to category Option[A] and Option[B] and the flatMap along with the f: A => Option[B] passed into it is supposed to define a Functor between them. But in the tranditional category definition it's a functor from a category to itself.

Answer 1

The category is the category of scala types, where the objects are types and the arrows are functions between values of those types. Option is an endofunctor on this category. For each object (ie type) in the Scala category, the Option type constructor maps each type A into a type Option[A] .

In addition it maps each arrow f: A => B into an arrow fo: Option[A] => Option[B] which is what Option.map does.

A Monad is a Functor M along with two operations, unit: A => M[A] and join: M[M[A]] => M[A] . For Option , unit(x: A) = Some(x) and join can be defined as:

def join[A](o: Option[Option[A]]): Option[A] = o match {
  case None => None
  case Some(i) => i
}

flatMap can then be defined as, flatMap(f, m) = join(map(f, m)) . Alternatively the monad can be defined using unit and flatMap and join defined as join(m) = flatMap(id, m) .